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Adaptive Binary Search Trees
, 2009
"... A ubiquitous problem in the field of algorithms and data structures is that of searching for an element from an ordered universe. The simple yet powerful binary search tree (BST) model provides a rich family of solutions to this problem. Although BSTs require Ω(lg n) time per operation in the wors ..."
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A ubiquitous problem in the field of algorithms and data structures is that of searching for an element from an ordered universe. The simple yet powerful binary search tree (BST) model provides a rich family of solutions to this problem. Although BSTs require Ω(lg n) time per operation in the worst case, various adaptive BST algorithms are capable of exploiting patterns in the sequence of queries to achieve tighter, inputsensitive, bounds that can be o(lg n) in many cases. This thesis furthers our understanding of what is achievable in the BST model along two directions. First, we make progress in improving instancespecific lower bounds in the BST model. In particular, we introduce a framework for generating lower bounds on the cost that any BST algorithm must pay to execute a query sequence,
Fast Local Searches and Updates in Bounded Universes
"... Given a bounded universe {0, 1,..., U −1}, we show how to perform (successor) searches in O(log log ∆) expected time and updates in O(log log ∆) expected amortized time, where ∆ is the rank difference between the element being searched for and its successor in the structure. This unifies the results ..."
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Given a bounded universe {0, 1,..., U −1}, we show how to perform (successor) searches in O(log log ∆) expected time and updates in O(log log ∆) expected amortized time, where ∆ is the rank difference between the element being searched for and its successor in the structure. This unifies the results of traditional bounded universe structures (which support successor searches in O(log log U) time) and hashing (which supports exact searches in O(1) time). We also show how these results can be extended to answer approximate nearest neighbour queries in low dimensions. 1
Research Statement
, 2010
"... Many “wellbehaved ” mathematical objects are not so wellbehaved in a computer system. A smooth function on a space may have nice mathematical structure but that does not mean we can represent it as bits. Many problems in robotics, machine learning, and data analysis are similar in that the underly ..."
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Many “wellbehaved ” mathematical objects are not so wellbehaved in a computer system. A smooth function on a space may have nice mathematical structure but that does not mean we can represent it as bits. Many problems in robotics, machine learning, and data analysis are similar in that the underlying object may be complex. Moreover, the inputs that stand as proxy for these objects are clouds of points in a geometric space. I work on algorithms and data structures for geometric problems on point clouds. My work strives to discover, describe, represent, and search this implicit structure. I mainly focus on methods to construct lowcomplexity simplicial complexes that accurately encode information about the input point set, underlying sample distribution, the ambient space, or some unknown function on the ambient space. My goal is to make sense of the complex, competing tradeoffs between the geometric, combinatorial, and topological information in important computational problems to produce useful, new algorithms. LowComplexity Complexes Throughout this research statement, let P be a set of n points in ddimensional Euclidean space. A funny thing happens when we build a simplicial complex P. The number of
ABSTRACT Title of dissertation: DYNAMIC DATA STRUCTURES FOR GEOMETRIC SEARCH AND RETRIEVAL
"... Data structures for geometric search and retrieval are of significant interest in areas such as computational geometry, computer graphics, and computer vision. The focus of this dissertation is on the design of efficient dynamic data structures for use in approximate retrieval problems in multidimen ..."
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Data structures for geometric search and retrieval are of significant interest in areas such as computational geometry, computer graphics, and computer vision. The focus of this dissertation is on the design of efficient dynamic data structures for use in approximate retrieval problems in multidimensional Euclidean space. A number of data structures have been proposed for storing multidimensional point sets. We will focus on quadtreebased structures. Quadtreebased structures possess a number of desirable properties, and they have been shown to be useful in solving a wide variety of query problems, especially when approximation is involved. First, we introduce two dynamic quadtreebased data structures for storing a set of points in space, called the quadtreap and the splay quadtree. The quadtreap is a randomized variant of a quadtree that supports insertion and deletion and has logarithmic height with high probability. The splay quadtree is also a quadtree variant, but this data structure is selfadjusting, that is, it rebalances itself depending on the access pattern. It supports efficient insertion and deletion in the amortized sense. We also study how to dynamically maintain an important geometric structure,